The Comprehensive Guide to Overtones and Harmonics: A Deep Dive into the Physics of Sound

Overtones and harmonics are fundamental concepts in the field of physics, particularly in acoustics and music. They refer to the additional frequencies that are present in a sound wave above and beyond the fundamental frequency, which play a crucial role in shaping the timbre and character of various musical instruments and natural sounds.

Understanding the Fundamentals

Defining Key Terms

1. Fundamental Frequency: The lowest frequency in a sound wave, which determines the perceived pitch.
2. Overtones: Any frequencies above the fundamental frequency, including harmonics and inharmonic frequencies.
3. Harmonics: Specific overtones that are integer multiples of the fundamental frequency.

The Harmonic Series

The harmonic series is a sequence of frequencies that are integer multiples of the fundamental frequency. For example, if the fundamental frequency is 100 Hz, the harmonic series would be:

Harmonic Number	Frequency (Hz)
1 (Fundamental)	100
2 (First Harmonic)	200
3 (Second Harmonic)	300
4 (Third Harmonic)	400
5 (Fourth Harmonic)	500

The ratio of the frequency of any harmonic to the fundamental frequency is a simple whole number, such as 2:1, 3:1, or 4:1. This is what gives harmonics their distinctive sound and makes them so important in music and acoustics.

Measuring Overtones and Harmonics

There are several key parameters used to describe and quantify overtones and harmonics:

1. Frequency: Measured in Hertz (Hz), the number of cycles per second that a sound wave completes.
2. Amplitude: Measured in decibels (dB), the height or magnitude of a sound wave.
3. Wavelength: Measured in meters (m), the distance between two consecutive points in a sound wave that are in phase with each other.
4. Phase: Measured in degrees or radians, the position of a point in a sound wave relative to a reference point.
5. Harmonic Series: The sequence of harmonics present in a sound wave, usually represented as a series of integer multiples of the fundamental frequency.
6. Overtones: Any frequencies above the fundamental frequency, including harmonics and inharmonic frequencies.
7. Inharmonicity: Measured in cents, the degree to which the overtones in a sound wave deviate from whole number ratios with the fundamental frequency.

Practical Applications of Overtones and Harmonics

Guitar String Vibration

When a guitar string is plucked and allowed to vibrate freely, it produces a sound wave that consists of the fundamental frequency and a series of harmonics. The frequencies of these harmonics can be calculated using the formula:

f_n = n * f_1

where f_n is the frequency of the nth harmonic, n is the harmonic number, and f_1 is the frequency of the first harmonic (i.e., the fundamental frequency).

For example, if the fundamental frequency of the guitar string is 100 Hz, the frequencies of the first five harmonics would be:

f_1 = 100 Hz
f_2 = 2 * f_1 = 200 Hz
f_3 = 3 * f_1 = 300 Hz
f_4 = 4 * f_1 = 400 Hz
f_5 = 5 * f_1 = 500 Hz

These harmonics are what give the guitar its distinctive sound and allow us to distinguish between different notes and chords.

Quartz Crystal Microbalances (QCMs)

Quartz crystal microbalances (QCMs) are used to measure the mass of thin films and other materials by detecting changes in the resonant frequency of a quartz crystal. By measuring the resonant frequency at multiple overtones, QCMs can provide more accurate and reliable measurements than if they only measured the fundamental frequency.

The number of allowed overtones in a QCM depends on the bandwidth of the instrument and the fundamental resonant frequency of the quartz. The overtone order must be odd (1, 3, 5, …) to ensure an antisymmetric pattern of motion of the quartz, which is necessary for accurate measurements.

Advanced Concepts and Numerical Examples

Inharmonicity and Partials

While harmonics are characterized by whole number ratios with the fundamental frequency, some sound sources, such as struck or plucked strings, can produce inharmonic overtones. These inharmonic overtones, also known as partials, do not follow the simple harmonic series and can contribute to the unique timbres of certain musical instruments.

The degree of inharmonicity can be quantified using the concept of inharmonicity, which is measured in cents. One cent is a logarithmic unit of measurement, where 100 cents is equal to one semitone in Western music. The inharmonicity of a sound wave can be calculated using the following formula:

I = (f_n – n * f_1) / (n * f_1) * 1200 cents

where I is the inharmonicity in cents, f_n is the frequency of the nth partial, n is the partial number, and f_1 is the fundamental frequency.

For example, consider a piano string with a fundamental frequency of 100 Hz. The first five partials might have the following frequencies:

f_1 = 100 Hz
f_2 = 201 Hz
f_3 = 303 Hz
f_4 = 406 Hz
f_5 = 510 Hz

Using the inharmonicity formula, we can calculate the degree of inharmonicity for each partial:

f_2: I = (201 – 2 * 100) / (2 * 100) * 1200 = 100 cents
f_3: I = (303 – 3 * 100) / (3 * 100) * 1200 = 67 cents
f_4: I = (406 – 4 * 100) / (4 * 100) * 1200 = 50 cents
f_5: I = (510 – 5 * 100) / (5 * 100) * 1200 = 40 cents

This inharmonicity contributes to the distinctive sound of the piano and other struck or plucked string instruments.

Numerical Example: Calculating Harmonics

Consider a guitar string with a fundamental frequency of 100 Hz. Calculate the frequencies of the first five harmonics.

Solution:
Using the formula f_n = n * f_1, where f_n is the frequency of the nth harmonic and f_1 is the fundamental frequency, we can calculate the frequencies of the first five harmonics:

f_1 = 100 Hz (Fundamental Frequency)
f_2 = 2 * f_1 = 2 * 100 Hz = 200 Hz (First Harmonic)
f_3 = 3 * f_1 = 3 * 100 Hz = 300 Hz (Second Harmonic)
f_4 = 4 * f_1 = 4 * 100 Hz = 400 Hz (Third Harmonic)
f_5 = 5 * f_1 = 5 * 100 Hz = 500 Hz (Fourth Harmonic)

Therefore, the frequencies of the first five harmonics are 100 Hz, 200 Hz, 300 Hz, 400 Hz, and 500 Hz.

Conclusion

Overtones and harmonics are fundamental concepts in the field of physics, with far-reaching applications in acoustics, music, and various scientific instruments. By understanding the properties and characteristics of these phenomena, we can gain deeper insights into the nature of sound and vibration, and unlock the secrets behind the rich and diverse soundscapes that surround us.

References

1. “Overtone.” Encyclopædia Britannica, 2024, www.britannica.com/science/sound-physics/Overtones.
2. “QCM: Why measure at overtones matters.” BioLogic Learning Center, 2024, www.biologic.net/topics/quartz-crystal-microbalance-why-measure-at-overtones/.
3. Reviakine, I., Morozov, A. N., & Rossetti, F. F. (2004). Fundamental Principles of Quartz Crystal Microbalance (QCM) Operation. Journal of Applied Physics, 95(11), 7712-7718.
4. “Fundamental Frequency and Harmonics.” The Physics Classroom, www.physicsclassroom.com/class/sound/Lesson-4/Fundamental-Frequency-and-Harmonics.
5. “How to compute fundamental frequency from a list of overtones?” DSP Stack Exchange, 2013, dsp.stackexchange.com/questions/8890/how-to-compute-fundamental-frequency-from-a-list-of-overtones.